We define the notion of a model of higher-order modal logic in an arbitrary elementary topos E . In contrast to the wellknown interpretation of (non-modal) higher-order logic, the type of propositions is not interpreted by the subobject classifier ΩE , but rather by a suitable complete Heyting algebra H . The canonical map relating H and ΩE both serves to interpret equality and provides a modal...

We introduce a new semantics for modal predicate logic, with respect to which a rich class of first-order modal logics is complete, namely all normal first-order modal logics that are extensions of free quantified K. This logic is defined by combining positive free logic with equality PFL . = and the propositional modal logic K. We then uniformly construct—for each modal predicate logic L—a can...

This paper extends the notion of prime implicates to first order logic formulas without equality which are assumed to be in Skolem Conjunctive Normal Form. Using the extended notions of consensus and subsumption it is shown that the consensus-subsumption algorithm for computing prime implicates well known for propositional formulas can be conditionally lifted to first order formulas.

This paper presents matching logic, a first-order logic (FOL) variant for specifying and reasoning about structure by means of patterns and pattern matching. Its sentences, the patterns, are constructed using variables, symbols, connectives and quantifiers, but no difference is made between function and predicate symbols. In models, a pattern evaluates into a power-set domain (the set of values...

A propositionalization of a theory in First-Order Logic (FOL) is a set of propositional sentences that is satisfiable iff the original theory is satisfiable. We cannot translate arbitrary FOL theories to propositional logic because FOL is only semi-decidable. However, when possible, it is often advantageous to do so because we can use optimized, efficient SAT solvers (e.g. [Moskewicz et al., 20...